Applied and Computational Mathematics 2016; 5(2): 64-72 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20160502.15 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation Sami H. Altoum 1 , Salih Y. Arbab 2 1 Department of Mathematics, University College of Qunfudha, Umm Alqura University, Makkah, KSA 2 Engineering College, Albaha University, Albaha, KSA Email address: [email protected] (S. H. Altoum), [email protected] (S. Y. Arbab) To cite this article: Sami H. Altoum, Salih Y. Arbab. Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation. Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 64-72. doi: 10.11648/j.acm.20160502.15 Received: February 17, 2016; Accepted: March 25, 2016; Published: April 15, 2016 Abstract: This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically. Keywords: Riccati Equation, Symmetry Group, Infinitesimal Generator, Runge-Kutta 1. Introduction ( ( ( 2 f(x,y) Px Qxy Rxy = + + (1) where ( ( ( P x ,Q x ,R x functions of x The Riccati differential equation is named after the Italian nobleman Count Jacopo Francesco Riccati (1676-1754). The book of Reid [1] contains the fundamental theories of Riccati equation. This equation is perhaps one of the simplest non- linear first ODE which plays a very important role in solution of various non-linear equations may be found in numerous scientific fields. The solution of this equation can be obtained numerically by using classical numerical method such as the forward Euler method and Runge-Kutta method. An analytic solution of the non-linear Riccati equation reached see [2] using A domain Decomposition Method. [5] used differential Transform Method(DTM) to solve Riccati differential equations with variable co-efficient and the results are compared with the numerical results by (RK4) method.[14] solved Riccati differential equations by using (ADM) method and the numerical results are compared with the exact solutions. B.Batiha [15] solved Riccati differential equations using Variational Iteration Method (VIM) and numerical comparison between VIM, RK4 and exact solution on these equations are given. Lie symmetries are utilized to solve both linear and non-linear first order ODE, and the majority of ad hoc methods of integration of ordinary differential equation could be explained and deduced simply by means of the theory of Lie group. Moreover Lie gave a classification of ordinary differential equation in terms of their symmetry groups. Lie's classification shows that the second order equations integrable by his method can be reduced to merely four distinct canonical forms by changes of variables. Subjecting these four canonical equations to changes of variables alone, one obtains all known equations integrated by classical methods as well as infinitely many unknown integrable equations. We will consider in our test only a first order differential equation and the numerical solution to second-order and higher-order differential equations is formulated in the same way as it is for first-order equations. We have attempted to give some examples to the use of Lie group methods for the solution of first-order ODEs. The Lie group method of solving higher order ODEs and systems of differential equations is more involved, but the basic idea is the same: we find a coordinate system in which the equations are simpler and exploit this simplification. Definition: A change of variables, ( ( x,y x, y → , is called an equivalence transformation of Riccati differential equation if any equation of the form (1) transformed into an
9
Embed
Comparison Solutions Between Lie Group Method and ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Applied and Computational Mathematics 2016; 5(2): 64-72
http://www.sciencepublishinggroup.com/j/acm
doi: 10.11648/j.acm.20160502.15
ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)
Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation
Sami H. Altoum1, Salih Y. Arbab
2
1Department of Mathematics, University College of Qunfudha, Umm Alqura University, Makkah, KSA
Example 3 Consider the Bernoulli equation ' 1 xy y y e−= + , when substituted into condition (10), leads to
( ) ( ) ( ) ( ) ( )( )21 x 1 x 1 x x 2
x y y xy y e y y e y e 1 e y 0− − −η − ξ + + η − ξ + − ξ + η − =
This, again, is too difficult as it sits, so we try a few
simplifying assumptions before we discover that
1, (y)ξ = η = η yields
( ) ( ) ( )1 x 1 x 2 x
y y y e y e 1 y e 0− − −η + + − η − =
Because some terms depend only on y , we solve
yy 0η − η = to obtain cyη = . Inserting this form of ´ into the
remaining equation 1
yy 1 0−η + η − = , we arrive at y 2η = .
Now that we have settled on the symbols ( ) ( ), 1, y 2ξ η = ,
we find canonical coordinates by solving dy y
dx 2
η= =ξ
to get
u and t. Remember that we seek families of functions that
remain constant for u so x 2u c ye .−= = The second
coordinate s is found by integrating dx
du1
= to get u x.=
The next step is to find the differential equation in the
canonical coordinates by computing;
x y
x y
u u hdu.
dt t t h
+=
+
We learn that
( )x 2 x 2 1 x x 2 1 x 2
du 1 1.
1 1dtye e y y e ye y e
2 2
− − − − −= =
+ + +
Expressing in x 2 1 x 21
ye y e2
− −+ terms of t and u leads to
t 1
2 t+ , whence
2
du t
dt t1
2
=+
This integrates to
2tu ln 1 c
2
= + +
Returning to the original coordinates, we obtain
2 xy ex ln 1 c
2
− = + +
( )x (x c)2e e 1 y− − =
Now from initial condition y(0.1) 2= we, get
( )0.1 0.1 c2e e 1 y− − =
0.1 c 0.14e 1 e
2
− =− =
0.1 c 0.1e 1 2e− −= +
c 0.1 0.2e e 2e− − −= +
( )0.1 0.2c ln e 2e− −= − +
( )0.2 0.1c ln e (2 e )−= − +
( )0.11c ln 2 e
5= − +
( )1
c 0.15e e 2 e−− = + .
Finally we, get
1
x x 0.15y 2e e e (2 e ) 1−
= + −
1
1 5 x 2x 2x 1 105
1 5
2 e e e 2e e e
ye
− + + =
Fig. 3(a). Solution RK4 and Lie Group case N=50.
71 Sami H. Altoum and Salih Y. Arbab: Comparison Solutions Between Lie Group Method and Numerical
Solution of (RK4) for Riccati Differential Equation
Fig. 3(b). Solution RK4 and Lie Group case N=200.
Fig. 3(c). Solution RK4 and Lie Group case N=300.
Fig. 3(d). Solution RK4 and Lie Group case N=900.
Fig. 3. Global Error.
Here, we compare the solution of the Bernoulli equation
obtained using Lie group and the numerical solution given by
RK4, see Figure 3(a). We see for even a moderate number of
steps, the agreement between the Runge- Kutta method and
the analytic solution is remarkable. We can quantify just how
much better the Runge-Kutta method does by defining a
measure of the global error e as the magnitude of the
discrepancy between the numerical and actual values of y(1),
see Figure 3(b). Clearly, the error for the Runge-Kutta
method is several orders of magnitude lower. Furthermore,
the global error curves both look linear on the log-log plot,
which suggests that there is a power law dependence of ε on
N. We can determine the power a by fitting a power law to
the data obtained in our Maple code this leads that the global
error using RK4 is ( ).3973507074O h this match our expectation
that the one step error is ( )5O h .
7. Conclusions
In this work we conclude that, firstly the general solution
of Riccati equation is facing the problem of (Trial and Error)
and it is lead to the classical method. Secondly we used
Numerical solution (Forward Euler, RK4) method and we
find the Lie group method is better than classical method but
it fail at the singularity. The numerical solution is accuracy
than symmetry group solution and classical method. In the
last, we note that the Euler method are tested numerically and
we have not convergence and the determined the global error
is about ( )0.9467945525O h and this don’t match our expectation.
The graphs in this have been performed using Maple 18.
References
[1] W. T. Reid, Riccati Differential Equations (Mathematics in science and engineering), New York: Academic Press, 1972.
Applied and Computational Mathematics 2016; 5(2): 64-72 72
[2] F. Dubois, A. Saidi, Unconditionally Stable Scheme for Riccati Equation, ESAIM Proceeding. 8(2000), 39-52.
[3] A. A. Bahnasawi, M. A. El-Tawil and A. Abdel-Naby, Solving Riccati Equation using Adomians Decomposition Method, App. Math. Comput. 157(2007), 503-514.
[4] T. Allahviraloo, Sh. S. Bahzadi. Application of Iterative Methods for Solving General Riccati Equation, Int. J. Industrial Mathematics, Vol. 4, ( 2012) No. 4, IJIM-00299.
[5] Supriya Mukherjee, Banamali Roy. Solution of Riccati Equation with Variable Co-efficient by Differential Transform Method, Int. J. of Nonlinear Science Vol.14, (2012) No.2, pp. 251-256.
[6] Taiwo, O. A., Osilagun J. A. Approximate Solution of Generalized Riccati Differential Equation by Iterative Decomposition Algorithm, International Journal of Engineering and Innivative Technology(IJEIT) Vol. 1(2012) No. 2, pp. 53-56.
[7] J. Biazar, M. Eslami. Differential Transform Method for Quadratic Riccati Differential Equation, vol. 9 (2010) No.4, pp. 444-447.
[8] Cristinel Mortici. The Method of the Variation of Constants for Riccati Equations, General Mathematics, Vol. 16(2008) No.1, pp. 111-116.
[9] B. Gbadamosi, O. adebimpe, E. I. Akinola, I. A. I. Olopade. Solving Riccati Equation using Adomian Decomposition Method, International Journal of Pure and Applied Mathematics, Vol. 78(2012) No. 3, pp. 409-417.
[10] Olever. P. J. Application of Lie Groups to Differential Equations. New York Springer-Verlag, (1993).
[11] Al Fred Grany. Modern Differential Geometry of Curves and Surfaces, CRC Press, (1998).
[12] Aubin Thierry. Differential Geometry, American Mathematical Society, (2001).
[13] Nail. H. Ibragimov, Elementry Lie Group Analysis and Ordinary Differential Equations, John Wiley Sons New York, (1996).
[14] T. R. Ramesh Rao, "The use of the A domain Decomposition Method for Solving Generalized Riccati Differential Equations" Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010) Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia pp. 935-941.
[15] B. Batiha, M. S. M. Noorani and I. Hashim, " Application of Variational Iteration Method to a General Riccati Equation" International Mathematical Forum, Vol.2, no. 56, pp. 2759–2770, 2007.